Theorem isgrpd2 13828

Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2238, but we make an exception for theorems such as isgrpd2 13828 and ismndd 13752 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
isgrpd2.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd2.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd2.z	⊢ (𝜑 → 0 = (0g‘𝐺))
isgrpd2.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
isgrpd2.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
isgrpd2.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd2	⊢ (𝜑 → 𝐺 ∈ Grp)

Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥

Allowed substitution hints:   𝑁(𝑥)   0 (𝑥)

Proof of Theorem isgrpd2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpd2.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		isgrpd2.p	. 2 ⊢ (𝜑 → + = (+g‘𝐺))
3		isgrpd2.z	. 2 ⊢ (𝜑 → 0 = (0g‘𝐺))
4		isgrpd2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
5		isgrpd2.n	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
6		isgrpd2.j	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
7		oveq1 6092	. . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
8	7	eqeq1d 2247	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
9	8	rspcev 2929	. . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
10	5, 6, 9	syl2anc 415	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
11	1, 2, 3, 4, 10	isgrpd2e 13827	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ‘cfv 5377  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  0_gc0g 13612  Mndcmnd 13731  Grpcgrp 13807

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-grp 13810

This theorem is used by:  prdsgrpd  14199